Search results for “complete intersections”

Every finite flat commutative group scheme over a noetherian local ring is the torsion component of the Picard scheme of a smooth projective scheme with 3-dimensional fibers, built as a quotient of a complete intersection. An application: Hodge numbers that jump in a smooth projective family.

A complete local ring with a finitely generated maximal ideal is Noetherian. Drop completeness and it fails: an ultraproduct construction gives a non-Noetherian local ring whose maximal ideal is principal.

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.

Real-time monitoring of scientific papers and technological news requires fast processing of complicated search demands motivated by thematically relevant information acquisition. For this case, the authors develop an exploratory search engine based on probabilistic hierarchical topic modeling. Topic model gives a low dimensional sparse interpretable vector representation (topical embedding) of a text, which is used for ranking documents by their similarity to the query. They explore several ways of comparing topical vectors including searching with thematically homogeneous text segments. Topical hierarchies are built using the regularized EM-algorithm from BigARTM project. The topic-based search achieves better precision and recall than other approaches (TF-IDF, fastText, LSTM, BERT) and even human assessors who spend up to an hour to complete the same search task. They also discover that blending hierarchical topic vectors with neural pretrained embeddings is a promising way of enriching both models that helps to get precision and recall higher than 90

A rigidity condition on schemes, strictly stronger than smoothness, introduced with examples and a look at where it fails to be geometric — no known application, just a curiosity.

An example, due to Serre, of a smooth projective surface in positive characteristic whose Picard scheme fails to be reduced, worked out via the two governing dimension inequalities.

A proof of Raynaud's theorem: every finite flat commutative group scheme over a locally noetherian base embeds, Zariski-locally, as a closed subgroup of an abelian scheme.

It is shown that computationally tight bounds for the probability of overfitting can be obtained only by simultaneous consideration of the following two properties of classifier sets: splitting into error levels and similarity of classifiers. For a set consisting of only two classifiers, an exact bound is obtained for the probability of overfitting. This is the simplest learning task that exhibits overfitting and the effects of splitting and similarity, which reduce the probability of overfitting. For a more complex case—a chain of classifiers—an experiment is carried out in which the effects of splitting and similarity are estimated separately. It is shown that reasonably low probabilities of overfitting can be obtained only for the sets that possess both properties.

This article is devoted to the problem of calculating an exact upper bound for the functionals of the generalization ability of a family of one-dimensional threshold decision rules. An algorithm is investigated that solves the stated problem and is polynomial in the total number of samples used for training and validation and in the number of training samples. A theorem is proved for calculating an estimate for the functional of expected overfitting and an estimate for the error rate of the method for minimizing empirical risk on a validation set. The exact bounds calculated using the theorem are compared with the previously known quick-to-compute upper bounds so as to estimate the orders of overestimation of the bounds and to identify the bounds that could be used in real problems.

This paper introduces an approach to measuring the cognitive complexity of texts on various language levels. While standard readability indices are based on the linear combination of primary statistics, our general approach allows us to estimate complexity on morphological, lexical, syntactic, and discursive levels. Each model is defined by the tokens for the specific language level and the complexity function of a single token. We then use the reference collection of moderately complex texts and the quantile-based approach to spot the abnormally rare tokens. The proposed supervised ensemble, based on the ElasticNet model, incorporates models from all language levels. Having collected a labeled dataset through crowdsourcing, consisting of pairs of articles from the Russian Wikipedia, we consider several models and ensembles and compare them to common baselines. Suggested models are flexible due to the freedom in choosing the reference collection. The described experiments confirm the competitiveness of the proposed approach, as the ensembles demonstrate the best target metric value.

A wide class of natural language processing tasks is solved using markup.At the moment, the vast majority of models and datasets rely on a simple markup structure containing only fragments and labels.Moreover, simple classification metrics such as F1, Precision, Recall are used to evaluate the model's accuracy.The problem with such metrics is that they do not take into account all aspects of the markup structure and that they are applicable only under the assumption of the existence of an ideal markup.This paper proposes a more general and universal markup structure that allows solving complex problems and builds a methodology for multi-criteria evaluation of text markup models based on inconsistent expert markup.After that, the application of the constructed method is considered to assess the quality of the model obtained within the winning algorithm of the "READ//ABLE" competition, which focused on building an effective essay markup system.The results demonstrate that the new markup structure and evaluation approach provides a more comprehensive and accurate assessment of model performance, addressing the limitations of traditional metrics by accounting for complex markup scenarios and expert inconsistencies.

Modern AI systems often comprise multiple learnable components that can be naturally organized as graphs. A central challenge is the end-to-end training of such systems without restrictive architectural or training assumptions. Such tasks fit the theory and approaches of the collaborative Multi-Agent Reinforcement Learning (MARL) field. We introduce Reinforcement Networks, a general framework for MARL that organizes agents as vertices in a directed acyclic graph (DAG). This structure extends hierarchical RL to arbitrary DAGs, enabling flexible credit assignment and scalable coordination while avoiding strict topologies, fully centralized training, and other limitations of current approaches. We formalize training and inference methods for the Reinforcement Networks framework and connect it to the LevelEnv concept to support reproducible construction, training, and evaluation. We demonstrate the effectiveness of our approach on several collaborative MARL setups by developing several Reinforcement Networks models that achieve improved performance over standard MARL baselines. Beyond empirical gains, Reinforcement Networks unify hierarchical, modular, and graph-structured views of MARL, opening a principled path toward designing and training complex multi-agent systems. We conclude with theoretical and practical directions - richer graph morphologies, compositional curricula, and graph-aware exploration. That positions Reinforcement Networks as a foundation for a new line of research in scalable, structured MARL.

The basic properties of addition, scalar multiplication, transposition, and multiplication of matrices, with justifications.

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.

Nonsingular matrices, the invertibility criterion, the adjugate-matrix formula, and the Gauss–Jordan method.

The axioms of a linear (vector) space, examples and counterexamples.

55th International Mathematical Olympiad. Cape Town, South Africa, 2014.

The Millennium Meeting, Paris, 24 May 2000, at which the seven problems were announced.